Density function for the GND with location parameter mu,
scale parameter sigma and shape parameter nu.
dgnd(x, mu = 0, sigma = 1, nu = 2)dgnd returns the density.
A numeric vector of observations.
A numeric value indicating the location parameter \(\mu\).
A numeric value indicating the scale parameter \(\sigma\).
A numeric value indicating the shape parameter \(\nu\).
If mu, sigma and nu are not specified
they assume the default values of 0, 1 and 2, respectively.
The GND distribution has density
$$ f_{GND}(x|\mu,\sigma,\nu)=\frac{\nu}{2\sigma\Gamma(1\mathbin{/}\nu)}\exp\Biggr\{-\Biggr|\frac{x-\mu}{\sigma}\Biggr|^\nu\Biggr\}.$$
The shape parameter \(\nu\) controls both the peakedness and tail weights.
If \(\nu=1\) the GND reduces to the Laplace distribution and if \(\nu=2\)
it coincides with the normal distribution. It is noticed that \(1<\nu<2\)
yields an intermediate distribution between the normal and the Laplace distribution.
As limit cases, for \(\nu\rightarrow\infty\) the distribution tends to a uniform
distribution, while for \(\nu\rightarrow0\) it will be impulsive.
Nadarajah, S. (2005). A generalized normal distribution. Journal of Applied Statistics, \(32(7):685–694\).